# Example: Plaque psoriasis HTA report

    library(multinma)
    options(mc.cores = parallel::detectCores())

    #> For execution on a local, multicore CPU with excess RAM we recommend calling
    #> options(mc.cores = parallel::detectCores())
    #>
    #> Attaching package: 'multinma'
    #> The following objects are masked from 'package:stats':
    #>
    #>     dgamma, pgamma, qgamma

This vignette describes the analysis of treatments for moderate-to-severe plaque psoriasis from an HTA report (Woolacott et al. 2006), replicating the analysis in NICE Technical Support Document 2 (Dias et al. 2011). The data are available in this package as `hta_psoriasis`:

    head(hta_psoriasis)
    #>   studyn   studyc year trtn             trtc sample_size PASI50 PASI75 PASI90
    #> 1      1  Elewski 2004    1  Supportive care         193     12      5      1
    #> 2      1  Elewski 2004    2 Etanercept 25 mg         196     59     46     21
    #> 3      1  Elewski 2004    3 Etanercept 50 mg         194     54     56     40
    #> 4      2 Gottlieb 2003    1  Supportive care          55      5      1      0
    #> 5      2 Gottlieb 2003    2 Etanercept 25 mg          57     23     11      6
    #> 6      3  Lebwohl 2003    1  Supportive care         122     13      5      1

Outcomes are ordered multinomial success/failure to achieve 50%, 75%, or 90% reduction in symptoms on the Psoriasis Area and Severity Index (PASI) scale. Some studies report ordered outcomes at all three cutpoints, others only one or two:

    dplyr::filter(hta_psoriasis, studyc %in% c("Elewski", "Gordon", "ACD2058g", "Altmeyer"))
    #>   studyn   studyc year trtn             trtc sample_size PASI50 PASI75 PASI90
    #> 1      1  Elewski 2004    1  Supportive care         193     12      5      1
    #> 2      1  Elewski 2004    2 Etanercept 25 mg         196     59     46     21
    #> 3      1  Elewski 2004    3 Etanercept 50 mg         194     54     56     40
    #> 4      5   Gordon 2003    1  Supportive care         187     18      8     NA
    #> 5      5   Gordon 2003    4       Efalizumab         369    118     98     NA
    #> 6      6 ACD2058g 2004    1  Supportive care         170     25     NA     NA
    #> 7      6 ACD2058g 2004    4       Efalizumab         162     99     NA     NA
    #> 8     10 Altmeyer 1994    1  Supportive care          51     NA      1     NA
    #> 9     10 Altmeyer 1994    6         Fumaderm          49     NA     12     NA

Here, the outcome counts are given as “exclusive” counts. That is, for a study reporting all outcomes (e.g. Elewski), the counts represent the categories 50 < PASI < 75, 75 < PASI < 90, and 90 < PASI < 100, and the corresponding columns are named by the lower end of the interval. Missing values are used where studies only report a subset of the outcomes. For a study reporting only two outcomes, say PASI50 and PASI75 as in Gordon, the counts represent the categories 50 < PASI < 75 and 75 < PASI < 100. For a study reporting only one outcome, say PASI70 as in Altmeyer, the count represents 70 < PASI < 100. We also need the count for the lowest category (i.e. no higher outcomes achieved), which is equal to the sample size minus the counts in the other observed categories.

### Setting up the network

We begin by setting up the network. We have arm-level ordered multinomial count data, so we use the function `set_agd_arm()`. The function `multi()` helps us to specify the ordered outcomes correctly.

    pso_net <- set_agd_arm(hta_psoriasis,
                           study = paste(studyc, year),
                           trt = trtc,
                           r = multi(r0 = sample_size - rowSums(cbind(PASI50, PASI75, PASI90), na.rm = TRUE),
                                     PASI50, PASI75, PASI90,
                                     inclusive = FALSE,
                                     type = "ordered"))
    pso_net
    #> A network with 16 AgD studies (arm-based).
    #>
    #> ------------------------------------------------------- AgD studies (arm-based) ----
    #>  Study         Treatments
    #>  ACD2058g 2004 2: Supportive care | Efalizumab
    #>  ACD2600g 2004 2: Supportive care | Efalizumab
    #>  Altmeyer 1994 2: Supportive care | Fumaderm
    #>  Chaudari 2001 2: Supportive care | Infliximab
    #>  Elewski 2004  3: Supportive care | Etanercept 25 mg | Etanercept 50 mg
    #>  Ellis 1991    2: Supportive care | Ciclosporin
    #>  Gordon 2003   2: Supportive care | Efalizumab
    #>  Gottlieb 2003 2: Supportive care | Etanercept 25 mg
    #>  Gottlieb 2004 2: Supportive care | Infliximab
    #>  Guenther 1991 2: Supportive care | Ciclosporin
    #>  ... plus 6 more studies
    #>
    #>  Outcome type: ordered (4 categories)
    #> ------------------------------------------------------------------------------------
    #> Total number of treatments: 8
    #> Total number of studies: 16
    #> Reference treatment is: Supportive care
    #> Network is connected

Plot the network structure.

    plot(pso_net, weight_edges = TRUE, weight_nodes = TRUE) +
      # Nudge the legend over
      ggplot2::theme(legend.box.spacing = ggplot2::unit(0.75, "in"),
                     plot.margin = ggplot2::margin(0.1, 0, 0.1, 0.75, "in"))

### Meta-analysis models

We fit both fixed effect (FE) and random effects (RE) models.

#### Fixed effect meta-analysis

First, we fit a fixed effect model using the `nma()` function with `trt_effects = "fixed"`, using a probit link function `link = "probit"`. We use \(\mathrm{N}(0, 10^2)\) prior distributions for the treatment effects \(d_k\), and \(\mathrm{N}(0, 100^2)\) prior distributions for the study-specific intercepts \(\mu_j\). We can examine the range of parameter values implied by these prior distributions with the `summary()` method:

    summary(normal(scale = 10))
    #> A Normal prior distribution: location = 0, scale = 10.
    #> 50% of the prior density lies between -6.74 and 6.74.
    #> 95% of the prior density lies between -19.6 and 19.6.
    summary(normal(scale = 100))
    #> A Normal prior distribution: location = 0, scale = 100.
    #> 50% of the prior density lies between -67.45 and 67.45.
    #> 95% of the prior density lies between -196 and 196.

We also need to specify prior distributions for the latent cutpoints \(c_\textrm{PASI75}\) and \(c_\textrm{PASI90}\) on the underlying scale - here the PASI standardised mean difference due to the probit link (the cutpoint \(c_\textrm{PASI50}=0\)). To make these easier to reason about, we actually specify priors on the differences between adjacent cutpoints, e.g. \(c_\textrm{PASI90} - c_\textrm{PASI75}\) and \(c_\textrm{PASI75} - c_\textrm{PASI50}\). These can be given any positive-valued prior distribution, and Stan will automatically impose the necessary ordering constraints behind the scenes. We choose to give these implicit flat priors `flat()`.

The model is fitted using the `nma()` function.

    pso_fit_FE <- nma(pso_net,
                      trt_effects = "fixed",
                      link = "probit",
                      prior_intercept = normal(scale = 100),
                      prior_trt = normal(scale = 10),
                      prior_aux = flat())
    #> Note: Setting "Supportive care" as the network reference treatment.

Basic parameter summaries are given by the `print()` method:

    pso_fit_FE
    #> A fixed effects NMA with a ordered likelihood (probit link).
    #> Inference for Stan model: ordered_multinomial.
    #> 4 chains, each with iter=2000; warmup=1000; thin=1;
    #> post-warmup draws per chain=1000, total post-warmup draws=4000.
    #>
    #>                         mean se_mean   sd     2.5%      25%      50%      75%    97.5% n_eff Rhat
    #> d[Ciclosporin]          1.91    0.01 0.34     1.29     1.67     1.89     2.13     2.60  1248    1
    #> d[Efalizumab]           1.19    0.00 0.06     1.08     1.15     1.19     1.23     1.30  1867    1
    #> d[Etanercept 25 mg]     1.51    0.00 0.10     1.32     1.45     1.51     1.57     1.70  2017    1
    #> d[Etanercept 50 mg]     1.92    0.00 0.10     1.73     1.85     1.92     1.98     2.12  2207    1
    #> d[Fumaderm]             1.48    0.01 0.47     0.64     1.16     1.45     1.78     2.49  2618    1
    #> d[Infliximab]           2.33    0.00 0.26     1.84     2.15     2.31     2.50     2.87  2920    1
    #> d[Methotrexate]         1.60    0.01 0.45     0.74     1.30     1.60     1.90     2.50  1639    1
    #> lp__                -3405.11    0.08 3.50 -3412.80 -3407.31 -3404.83 -3402.62 -3399.14  1785    1
    #> cc[PASI50]              0.00     NaN 0.00     0.00     0.00     0.00     0.00     0.00   NaN  NaN
    #> cc[PASI75]              0.76    0.00 0.03     0.70     0.74     0.76     0.78     0.82  5459    1
    #> cc[PASI90]              1.57    0.00 0.05     1.46     1.53     1.57     1.60     1.67  6028    1
    #>
    #> Samples were drawn using NUTS(diag_e) at Fri Feb 19 11:40:28 2021.
    #> For each parameter, n_eff is a crude measure of effective sample size,
    #> and Rhat is the potential scale reduction factor on split chains (at
    #> convergence, Rhat=1).

> Note: the treatment effects are the opposite sign to those in TSD 2 (Dias et al. 2011). This is because we parameterise the linear predictor as \(\mu_j + d_k + c_m\), rather than \(\mu_j + d_k - c_m\). The interpretation here thus follows that of a standard binomial probit (or logit) regression; SMDs (or log ORs) greater than zero mean that the treatment increases the probability of an event compared to the comparator (and less than zero mean a reduction in probability). Here higher outcomes are positive, and all of the active treatments are estimated to increase the response (i.e. a greater reduction) on the PASI scale compared to the network reference (supportive care).

By default, summaries of the study-specific intercepts \(\mu_j\) are hidden, but could be examined by changing the `pars` argument:

    # Not run
    print(pso_fit_FE, pars = c("d", "mu", "cc"))

The prior and posterior distributions can be compared visually using the `plot_prior_posterior()` function:

    plot_prior_posterior(pso_fit_FE)

Focusing specifically on the cutpoints we see that these are highly identified by the data, which is why the implicit flat priors work for these parameters.

    plot_prior_posterior(pso_fit_FE, prior = "aux")

#### Random effects meta-analysis

We now fit a random effects model using the `nma()` function with `trt_effects = "random"`. Again, we use \(\mathrm{N}(0, 10^2)\) prior distributions for the treatment effects \(d_k\), \(\mathrm{N}(0, 100^2)\) prior distributions for the study-specific intercepts \(\mu_j\), implicit flat prior distributions for the latent cutpoints, and we additionally use a \(\textrm{half-N}(2.5^2)\) prior for the heterogeneity standard deviation \(\tau\). We can examine the range of parameter values implied by these prior distributions with the `summary()` method:

    summary(normal(scale = 10))
    #> A Normal prior distribution: location = 0, scale = 10.
    #> 50% of the prior density lies between -6.74 and 6.74.
    #> 95% of the prior density lies between -19.6 and 19.6.
    summary(normal(scale = 100))
    #> A Normal prior distribution: location = 0, scale = 100.
    #> 50% of the prior density lies between -67.45 and 67.45.
    #> 95% of the prior density lies between -196 and 196.
    summary(half_normal(scale = 2.5))
    #> A half-Normal prior distribution: location = 0, scale = 2.5.
    #> 50% of the prior density lies between 0 and 1.69.
    #> 95% of the prior density lies between 0 and 4.9.

Fitting the RE model

    pso_fit_RE <- nma(pso_net,
                      trt_effects = "random",
                      link = "probit",
                      prior_intercept = normal(scale = 100),
                      prior_trt = normal(scale = 10),
                      prior_aux = flat(),
                      prior_het = half_normal(scale = 2.5),
                      adapt_delta = 0.99)

    #> Note: Setting "Supportive care" as the network reference treatment.

Basic parameter summaries are given by the `print()` method:

    pso_fit_RE
    #> A random effects NMA with a ordered likelihood (probit link).
    #> Inference for Stan model: ordered_multinomial.
    #> 4 chains, each with iter=2000; warmup=1000; thin=1;
    #> post-warmup draws per chain=1000, total post-warmup draws=4000.
    #>
    #>                         mean se_mean   sd     2.5%      25%      50%      75%    97.5% n_eff Rhat
    #> d[Ciclosporin]          2.04    0.01 0.45     1.29     1.73     2.01     2.30     3.05  1073 1.00
    #> d[Efalizumab]           1.19    0.00 0.19     0.77     1.10     1.19     1.28     1.56  1495 1.00
    #> d[Etanercept 25 mg]     1.52    0.01 0.26     0.99     1.40     1.52     1.65     2.06  1424 1.00
    #> d[Etanercept 50 mg]     1.93    0.01 0.29     1.33     1.79     1.92     2.06     2.53  1123 1.01
    #> d[Fumaderm]             1.49    0.01 0.63     0.36     1.08     1.46     1.87     2.80  3232 1.00
    #> d[Infliximab]           2.32    0.01 0.40     1.51     2.09     2.32     2.56     3.06  3707 1.00
    #> d[Methotrexate]         1.72    0.02 0.64     0.61     1.31     1.69     2.08     3.06  1502 1.00
    #> lp__                -3410.70    0.31 6.74 -3424.14 -3415.44 -3410.47 -3405.94 -3398.19   481 1.00
    #> tau                     0.31    0.01 0.23     0.02     0.15     0.27     0.42     0.91   317 1.01
    #> cc[PASI50]              0.00     NaN 0.00     0.00     0.00     0.00     0.00     0.00   NaN  NaN
    #> cc[PASI75]              0.76    0.00 0.03     0.70     0.74     0.76     0.78     0.82 10176 1.00
    #> cc[PASI90]              1.56    0.00 0.05     1.47     1.53     1.56     1.60     1.66  9324 1.00
    #>
    #> Samples were drawn using NUTS(diag_e) at Fri Feb 19 11:42:18 2021.
    #> For each parameter, n_eff is a crude measure of effective sample size,
    #> and Rhat is the potential scale reduction factor on split chains (at
    #> convergence, Rhat=1).

By default, summaries of the study-specific intercepts \(\mu_j\) and study-specific relative effects \(\delta_{jk}\) are hidden, but could be examined by changing the `pars` argument:

    # Not run
    print(pso_fit_RE, pars = c("d", "cc", "mu", "delta"))

The prior and posterior distributions can be compared visually using the `plot_prior_posterior()` function:

    plot_prior_posterior(pso_fit_RE, prior = c("trt", "aux", "het"))

#### Model comparison

Model fit can be checked using the `dic()` function:

    (dic_FE <- dic(pso_fit_FE))
    #> Residual deviance: 74.8 (on 58 data points)
    #>                pD: 25.4
    #>               DIC: 100.2

    (dic_RE <- dic(pso_fit_RE))
    #> Residual deviance: 63.1 (on 58 data points)
    #>                pD: 33.8
    #>               DIC: 96.9

The random effects model has a lower DIC and the residual deviance is closer to the number of data points, so is preferred in this case.

We can also examine the residual deviance contributions with the corresponding `plot()` method.

    plot(dic_FE)

    plot(dic_RE)

Most data points are fit well, with posterior mean residual deviances close to the degrees of freedom. The Meffert 1997 study has a substantially higher residual deviance contribution, which could be investigated further to see why this study appears to be an outlier.

### Further results

#### Predicted probabilities of response

Dias et al. (2011) produce absolute predictions of probability of achieving responses at each PASI cutoff, assuming a Normal distribution for the baseline probit probability of PASI50 response on supportive care with mean \(-1.097\) and precision \(123\). We can replicate these results using the `predict()` method. The `baseline` argument takes a `distr()` distribution object, with which we specify the corresponding Normal distribution. We set `type = "response"` to produce predicted probabilities (`type = "link"` would produce predicted probit probabilities).

    pred_FE <- predict(pso_fit_FE,
                       baseline = distr(qnorm, mean = -1.097, sd = 123^-0.5),
                       type = "response")
    pred_FE
    #>                                mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> pred[Supportive care, PASI50]  0.14 0.02 0.10 0.12 0.14 0.15  0.18     3675     3431    1
    #> pred[Supportive care, PASI75]  0.03 0.01 0.02 0.03 0.03 0.04  0.05     3700     3564    1
    #> pred[Supportive care, PASI90]  0.00 0.00 0.00 0.00 0.00 0.00  0.01     4056     3843    1
    #> pred[Ciclosporin, PASI50]      0.78 0.10 0.57 0.71 0.79 0.85  0.94     1418     1844    1
    #> pred[Ciclosporin, PASI75]      0.52 0.13 0.28 0.42 0.52 0.61  0.78     1422     1848    1
    #> pred[Ciclosporin, PASI90]      0.24 0.11 0.08 0.16 0.22 0.30  0.49     1432     1850    1
    #> pred[Efalizumab, PASI50]       0.54 0.04 0.45 0.51 0.54 0.57  0.62     3034     3365    1
    #> pred[Efalizumab, PASI75]       0.25 0.04 0.19 0.23 0.25 0.28  0.33     3039     3542    1
    #> pred[Efalizumab, PASI90]       0.07 0.02 0.04 0.06 0.07 0.08  0.11     2916     3577    1
    #> pred[Etanercept 25 mg, PASI50] 0.66 0.05 0.56 0.63 0.66 0.69  0.75     2770     3324    1
    #> pred[Etanercept 25 mg, PASI75] 0.37 0.05 0.27 0.33 0.36 0.40  0.47     2816     2892    1
    #> pred[Etanercept 25 mg, PASI90] 0.13 0.03 0.08 0.11 0.12 0.14  0.19     2846     3012    1
    #> pred[Etanercept 50 mg, PASI50] 0.79 0.04 0.71 0.77 0.79 0.82  0.86     2903     3108    1
    #> pred[Etanercept 50 mg, PASI75] 0.53 0.05 0.42 0.49 0.53 0.56  0.63     2930     2763    1
    #> pred[Etanercept 50 mg, PASI90] 0.23 0.04 0.16 0.20 0.23 0.26  0.32     2939     3201    1
    #> pred[Fumaderm, PASI50]         0.63 0.16 0.32 0.52 0.64 0.76  0.92     2834     2182    1
    #> pred[Fumaderm, PASI75]         0.37 0.17 0.11 0.24 0.34 0.48  0.74     2845     2208    1
    #> pred[Fumaderm, PASI90]         0.14 0.11 0.02 0.06 0.11 0.19  0.43     2856     2170    1
    #> pred[Infliximab, PASI50]       0.88 0.05 0.76 0.85 0.89 0.92  0.96     2908     2788    1
    #> pred[Infliximab, PASI75]       0.68 0.10 0.48 0.61 0.68 0.74  0.85     2952     2780    1
    #> pred[Infliximab, PASI90]       0.37 0.10 0.19 0.30 0.37 0.44  0.60     2987     2907    1
    #> pred[Methotrexate, PASI50]     0.68 0.15 0.36 0.58 0.69 0.79  0.92     1724     2233    1
    #> pred[Methotrexate, PASI75]     0.41 0.16 0.13 0.29 0.40 0.52  0.74     1723     2315    1
    #> pred[Methotrexate, PASI90]     0.17 0.11 0.03 0.09 0.14 0.22  0.44     1739     2261    1
    plot(pred_FE)

    pred_RE <- predict(pso_fit_RE,
                       baseline = distr(qnorm, mean = -1.097, sd = 123^-0.5),
                       type = "response")
    pred_RE
    #>                                mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> pred[Supportive care, PASI50]  0.14 0.02 0.10 0.12 0.14 0.15  0.18     4023     3849    1
    #> pred[Supportive care, PASI75]  0.03 0.01 0.02 0.03 0.03 0.04  0.05     4429     3935    1
    #> pred[Supportive care, PASI90]  0.00 0.00 0.00 0.00 0.00 0.00  0.01     4860     4105    1
    #> pred[Ciclosporin, PASI50]      0.80 0.11 0.56 0.74 0.82 0.88  0.98     1415      897    1
    #> pred[Ciclosporin, PASI75]      0.56 0.16 0.28 0.45 0.56 0.67  0.89     1413      893    1
    #> pred[Ciclosporin, PASI90]      0.28 0.15 0.08 0.17 0.26 0.36  0.65     1414      868    1
    #> pred[Efalizumab, PASI50]       0.53 0.08 0.36 0.49 0.54 0.58  0.69     2060     1302    1
    #> pred[Efalizumab, PASI75]       0.26 0.07 0.13 0.22 0.25 0.29  0.40     2090     1270    1
    #> pred[Efalizumab, PASI90]       0.07 0.03 0.03 0.06 0.07 0.09  0.14     2143     1414    1
    #> pred[Etanercept 25 mg, PASI50] 0.66 0.09 0.45 0.61 0.66 0.71  0.84     2002     1032    1
    #> pred[Etanercept 25 mg, PASI75] 0.37 0.10 0.19 0.32 0.37 0.42  0.59     2036     1054    1
    #> pred[Etanercept 25 mg, PASI90] 0.14 0.06 0.05 0.10 0.13 0.16  0.28     2065     1066    1
    #> pred[Etanercept 50 mg, PASI50] 0.79 0.09 0.58 0.75 0.80 0.84  0.92     1614      877    1
    #> pred[Etanercept 50 mg, PASI75] 0.53 0.11 0.29 0.47 0.53 0.59  0.75     1594      793    1
    #> pred[Etanercept 50 mg, PASI90] 0.24 0.09 0.09 0.19 0.23 0.28  0.45     1566      864    1
    #> pred[Fumaderm, PASI50]         0.63 0.20 0.22 0.49 0.64 0.78  0.96     3576     2090    1
    #> pred[Fumaderm, PASI75]         0.38 0.20 0.06 0.22 0.34 0.51  0.83     3603     2142    1
    #> pred[Fumaderm, PASI90]         0.16 0.15 0.01 0.06 0.11 0.22  0.56     3599     2020    1
    #> pred[Infliximab, PASI50]       0.87 0.08 0.66 0.84 0.89 0.93  0.98     3641     2308    1
    #> pred[Infliximab, PASI75]       0.67 0.14 0.36 0.59 0.68 0.76  0.89     3634     2332    1
    #> pred[Infliximab, PASI90]       0.37 0.14 0.13 0.28 0.37 0.46  0.67     3658     2420    1
    #> pred[Methotrexate, PASI50]     0.70 0.18 0.30 0.58 0.72 0.84  0.98     1955     1070    1
    #> pred[Methotrexate, PASI75]     0.45 0.21 0.10 0.29 0.43 0.60  0.89     1962     1084    1
    #> pred[Methotrexate, PASI90]     0.21 0.17 0.02 0.09 0.16 0.28  0.67     1960     1093    1
    plot(pred_RE)

If instead of information on the baseline PASI 50 response probit probability we have PASI 50 event counts, we can use these to construct a Beta distribution for the baseline probability of PASI 50 response. For example, if 56 out of 408 individuals achieved PASI 50 response on supportive care in the target population of interest, the appropriate Beta distribution for the response probability would be \(\textrm{Beta}(56, 408-56)\). We can specify this Beta distribution for the baseline response using the `baseline_type = "reponse"` argument (the default is `"link"`, used above for the baseline probit probability).

    pred_FE_beta <- predict(pso_fit_FE,
                            baseline = distr(qbeta, 56, 408-56),
                            baseline_type = "response",
                            type = "response")
    pred_FE_beta
    #>                                mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> pred[Supportive care, PASI50]  0.14 0.02 0.11 0.13 0.14 0.15  0.17     4044     3959    1
    #> pred[Supportive care, PASI75]  0.03 0.01 0.02 0.03 0.03 0.04  0.05     4048     3860    1
    #> pred[Supportive care, PASI90]  0.00 0.00 0.00 0.00 0.00 0.00  0.01     4507     3884    1
    #> pred[Ciclosporin, PASI50]      0.78 0.10 0.57 0.72 0.79 0.85  0.94     1373     1898    1
    #> pred[Ciclosporin, PASI75]      0.52 0.13 0.28 0.43 0.51 0.61  0.78     1374     1855    1
    #> pred[Ciclosporin, PASI90]      0.24 0.11 0.08 0.16 0.22 0.30  0.49     1395     1949    1
    #> pred[Efalizumab, PASI50]       0.54 0.04 0.46 0.51 0.54 0.56  0.62     3037     3686    1
    #> pred[Efalizumab, PASI75]       0.26 0.03 0.20 0.23 0.25 0.28  0.32     2995     3379    1
    #> pred[Efalizumab, PASI90]       0.07 0.02 0.05 0.06 0.07 0.08  0.10     3277     3741    1
    #> pred[Etanercept 25 mg, PASI50] 0.66 0.05 0.57 0.63 0.66 0.69  0.75     2507     3049    1
    #> pred[Etanercept 25 mg, PASI75] 0.37 0.05 0.28 0.33 0.37 0.40  0.47     2533     3312    1
    #> pred[Etanercept 25 mg, PASI90] 0.13 0.03 0.08 0.11 0.13 0.14  0.19     2693     3128    1
    #> pred[Etanercept 50 mg, PASI50] 0.79 0.04 0.72 0.77 0.79 0.82  0.86     2573     3247    1
    #> pred[Etanercept 50 mg, PASI75] 0.53 0.05 0.43 0.49 0.53 0.56  0.63     2533     3153    1
    #> pred[Etanercept 50 mg, PASI90] 0.23 0.04 0.16 0.20 0.23 0.26  0.31     2649     3277    1
    #> pred[Fumaderm, PASI50]         0.63 0.16 0.32 0.52 0.64 0.75  0.92     2827     2222    1
    #> pred[Fumaderm, PASI75]         0.37 0.17 0.11 0.24 0.34 0.47  0.74     2834     2222    1
    #> pred[Fumaderm, PASI90]         0.14 0.11 0.02 0.07 0.11 0.19  0.43     2848     2289    1
    #> pred[Infliximab, PASI50]       0.88 0.05 0.77 0.85 0.89 0.92  0.96     3009     2988    1
    #> pred[Infliximab, PASI75]       0.68 0.10 0.49 0.61 0.68 0.75  0.85     3048     2964    1
    #> pred[Infliximab, PASI90]       0.37 0.10 0.20 0.30 0.37 0.44  0.60     3106     2986    1
    #> pred[Methotrexate, PASI50]     0.68 0.15 0.36 0.58 0.69 0.79  0.92     1707     2292    1
    #> pred[Methotrexate, PASI75]     0.41 0.16 0.13 0.29 0.40 0.52  0.74     1702     2215    1
    #> pred[Methotrexate, PASI90]     0.17 0.11 0.03 0.08 0.14 0.23  0.45     1723     2182    1
    plot(pred_FE_beta)

    pred_RE_beta <- predict(pso_fit_RE,
                            baseline = distr(qbeta, 56, 408-56),
                            baseline_type = "response",
                            type = "response")
    pred_RE_beta
    #>                                mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> pred[Supportive care, PASI50]  0.14 0.02 0.11 0.13 0.14 0.15  0.17     4108     3777    1
    #> pred[Supportive care, PASI75]  0.03 0.01 0.02 0.03 0.03 0.04  0.05     4579     3925    1
    #> pred[Supportive care, PASI90]  0.00 0.00 0.00 0.00 0.00 0.00  0.01     5031     3914    1
    #> pred[Ciclosporin, PASI50]      0.81 0.11 0.57 0.74 0.82 0.89  0.97     1394      911    1
    #> pred[Ciclosporin, PASI75]      0.57 0.16 0.28 0.45 0.56 0.67  0.88     1393      976    1
    #> pred[Ciclosporin, PASI90]      0.28 0.15 0.08 0.18 0.26 0.36  0.65     1392      876    1
    #> pred[Efalizumab, PASI50]       0.54 0.08 0.36 0.49 0.54 0.58  0.69     2181     1368    1
    #> pred[Efalizumab, PASI75]       0.26 0.07 0.13 0.22 0.25 0.29  0.40     2224     1377    1
    #> pred[Efalizumab, PASI90]       0.07 0.03 0.03 0.06 0.07 0.09  0.14     2273     1397    1
    #> pred[Etanercept 25 mg, PASI50] 0.66 0.09 0.45 0.61 0.66 0.72  0.84     2040     1056    1
    #> pred[Etanercept 25 mg, PASI75] 0.37 0.10 0.19 0.32 0.37 0.42  0.59     2056     1063    1
    #> pred[Etanercept 25 mg, PASI90] 0.14 0.06 0.05 0.10 0.13 0.16  0.28     2085     1025    1
    #> pred[Etanercept 50 mg, PASI50] 0.79 0.08 0.59 0.75 0.80 0.84  0.92     1712      894    1
    #> pred[Etanercept 50 mg, PASI75] 0.53 0.11 0.30 0.47 0.53 0.59  0.75     1721      886    1
    #> pred[Etanercept 50 mg, PASI90] 0.24 0.09 0.09 0.19 0.23 0.28  0.45     1791      973    1
    #> pred[Fumaderm, PASI50]         0.63 0.20 0.23 0.49 0.64 0.78  0.96     3615     2080    1
    #> pred[Fumaderm, PASI75]         0.38 0.20 0.07 0.22 0.35 0.51  0.83     3635     2174    1
    #> pred[Fumaderm, PASI90]         0.16 0.15 0.01 0.06 0.11 0.21  0.56     3625     2009    1
    #> pred[Infliximab, PASI50]       0.87 0.08 0.66 0.84 0.89 0.93  0.98     3718     1943    1
    #> pred[Infliximab, PASI75]       0.67 0.13 0.36 0.59 0.68 0.76  0.89     3716     1988    1
    #> pred[Infliximab, PASI90]       0.37 0.14 0.12 0.28 0.37 0.46  0.66     3737     1949    1
    #> pred[Methotrexate, PASI50]     0.70 0.18 0.30 0.59 0.72 0.84  0.98     1952     1098    1
    #> pred[Methotrexate, PASI75]     0.45 0.21 0.10 0.29 0.43 0.59  0.89     1960     1049    1
    #> pred[Methotrexate, PASI90]     0.21 0.17 0.02 0.09 0.16 0.28  0.66     1964     1026    1
    plot(pred_RE_beta)

(Notice that these results are equivalent to those calculated above using the Normal distribution for the baseline probit probability, since these event counts correspond to the same probit probability.)

We can modify the plots using standard `ggplot2` functions. For example, to plot the cutpoints together with a colour coding (instead of split into facets):

    library(ggplot2)
    plot(pred_RE, position = position_dodge(width = 0.75)) +
      facet_null() +
      aes(colour = Category) +
      scale_colour_brewer(palette = "Blues")

If the `baseline` argument is omitted, predicted probabilities will be produced for every study in the network based on their estimated baseline probit probability \(\mu_j\).

#### Ranks and rank probabilities

Treatment rankings, rank probabilities, and cumulative rank probabilities can also be produced. We set `lower_better = FALSE` since higher outcome categories are better (the outcomes are positive).

    (pso_ranks <- posterior_ranks(pso_fit_RE, lower_better = FALSE))
    #>                        mean   sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> rank[Supportive care]  7.99 0.12    8   8   8   8     8     1346       NA    1
    #> rank[Ciclosporin]      2.74 1.28    1   2   3   4     5     3404     3047    1
    #> rank[Efalizumab]       6.32 0.83    4   6   6   7     7     2290     2014    1
    #> rank[Etanercept 25 mg] 4.94 1.08    3   4   5   6     7     2962     3061    1
    #> rank[Etanercept 50 mg] 3.04 1.22    1   2   3   4     6     1990     1868    1
    #> rank[Fumaderm]         4.91 1.95    1   3   5   7     7     3486     1253    1
    #> rank[Infliximab]       1.84 1.23    1   1   1   2     5     1327     1624    1
    #> rank[Methotrexate]     4.22 1.87    1   3   4   6     7     2792     2651    1
    plot(pso_ranks)

    (pso_rankprobs <- posterior_rank_probs(pso_fit_RE, lower_better = FALSE))
    #>                     p_rank[1] p_rank[2] p_rank[3] p_rank[4] p_rank[5] p_rank[6] p_rank[7] p_rank[8]
    #> d[Supportive care]       0.00      0.00      0.00      0.00      0.00      0.00      0.01      0.99
    #> d[Ciclosporin]           0.18      0.30      0.26      0.17      0.08      0.02      0.00      0.00
    #> d[Efalizumab]            0.00      0.00      0.01      0.02      0.10      0.36      0.50      0.00
    #> d[Etanercept 25 mg]      0.00      0.02      0.09      0.20      0.39      0.26      0.05      0.00
    #> d[Etanercept 50 mg]      0.09      0.29      0.28      0.24      0.09      0.02      0.01      0.00
    #> d[Fumaderm]              0.07      0.09      0.10      0.12      0.16      0.19      0.28      0.01
    #> d[Infliximab]            0.58      0.18      0.13      0.07      0.03      0.01      0.00      0.00
    #> d[Methotrexate]          0.09      0.13      0.15      0.19      0.16      0.14      0.15      0.00
    plot(pso_rankprobs)

    (pso_cumrankprobs <- posterior_rank_probs(pso_fit_RE, lower_better = FALSE, cumulative = TRUE))
    #>                     p_rank[1] p_rank[2] p_rank[3] p_rank[4] p_rank[5] p_rank[6] p_rank[7] p_rank[8]
    #> d[Supportive care]       0.00      0.00      0.00      0.00      0.00      0.00      0.01         1
    #> d[Ciclosporin]           0.18      0.47      0.73      0.90      0.98      1.00      1.00         1
    #> d[Efalizumab]            0.00      0.00      0.01      0.03      0.14      0.50      1.00         1
    #> d[Etanercept 25 mg]      0.00      0.02      0.10      0.30      0.69      0.95      1.00         1
    #> d[Etanercept 50 mg]      0.09      0.37      0.65      0.89      0.97      0.99      1.00         1
    #> d[Fumaderm]              0.07      0.16      0.26      0.37      0.53      0.71      0.99         1
    #> d[Infliximab]            0.58      0.76      0.89      0.96      0.98      1.00      1.00         1
    #> d[Methotrexate]          0.09      0.22      0.36      0.55      0.71      0.85      1.00         1
    plot(pso_cumrankprobs)

## References

Dias, S., N. J. Welton, A. J. Sutton, and A. E. Ades. 2011. “NICE DSU Technical Support Document 2: A Generalised Linear Modelling Framework for Pair-Wise and Network Meta-Analysis of Randomised Controlled Trials.” National Institute for Health and Care Excellence. http://nicedsu.org.uk/.
Woolacott, N., N. Hawkins, A. Mason, A. Kainth, Z. Khadjesari, Y. Bravo Vergel, K. Misso, et al. 2006. “Etanercept and Efalizumab for the Treatment of Psoriasis: A Systematic Review.” Health Technology Assessment 10 (46). https://doi.org/10.3310/hta10460.